Let's go through each step of Tyrianne's work to identify where an error might have occurred.
1. [tex]\(\frac{1}{2}(x+4)^2 - 3 = 29\)[/tex]
This is the original equation given. There is no mistake here.
2. [tex]\(\frac{1}{2}(x+4)^2 = 32\)[/tex]
Tyrianne added 3 to both sides to isolate the quadratic term. This step is correct.
3. [tex]\((x+4)^2 = 64\)[/tex]
Tyrianne multiplied both sides by 2 to eliminate the fraction. This step is also correct.
4. [tex]\(x+4 = \pm 8\)[/tex]
Tyrianne took the square root of both sides. Taking the square root of 64 correctly gives [tex]\(\pm 8\)[/tex]. This step is correct.
5. [tex]\(x = 8 - 4 \text{ or } x = -8 - 4\)[/tex]
Therefore, the solutions should be [tex]\(x = 4 \text{ or } x = -12\)[/tex]. Tyrianne incorrectly simplified the equation to [tex]\(x = 0 \text{ or } x = -8\)[/tex] instead of [tex]\(x = 4 \text{ or } x = -12\)[/tex].
The mistake was made in Step 5.
So, Tyrianne made an error in Step 5.
